OPEN SETS OF MAXIMAL DIMENSION IN COMPLEX HYPERBOLIC
QUASI-FUCHSIAN SPACE
J. R. PARKER & I. D. PLATIS
Abstract. Let π1 be the fundamental group of a closed surface Σ of genus g > 1. One of the
fundamental problems in complex hyperbolic geometry is to find all discrete, faithful, geometrically
finite and purely loxodromic representations of π1 into SU(2, 1), (the triple cover of) the group
of holomorphic isometries of H2C . In particular, given a discrete, faithful, geometrically finite and
purely loxodromic representation ρ0 of π1 , can we find an open neighbourhood of ρ0 comprising
representations with these properties. We show that this is indeed the case when ρ0 preserves a
totally real Lagrangian plane.
1. Introduction
Let Σ be a closed surface of genus g > 1 and let π1 = π1 (Σ) denote its fundamental group. A
specific choice of generators for π1 is called a marking. The collection of marked representations of
π1 into a Lie group G up to conjugation will be denote Hom(π1 , G)/G. We give Hom(π1 , G)/G the
compact-open topology. This enables us to make sense of what it means for two representations
to be close. In the cases we consider, the compact-open topology is equivalent to the l2 -topology
on the relevant matrix group. Our main interest in this paper will be the case where G = SU(2, 1)
but, before we consider this case, we motivate our discussion by reviewing the better known cases
when G is SL(2, R) or SL(2, C).
Suppose that ρ : π1 −→ SL(2, R) is a discrete and faithful representation of π1 . Then ρ(π1 )
is called Fuchsian. Also, ρ(π1 ) is necessarily geometrically finite and totally loxodromic (if Σ
had punctures then this condition would be replaced with type-preserving, which requires that an
element of ρ(π1 ) is parabolic if and only if it represents a peripheral curve). The group SL(2, R)
is a double cover of the group of orientation preserving isometries of the hyperbolic plane. The
quotient of the hyperbolic plane by ρ(π1 ) naturally corresponds to a hyperbolic structure on Σ.
The collection of distinct, marked Fuchsian representations, up to conjugacy within SL(2, R), is the
Teichmüller space of Σ, denoted T = T (Σ) ⊂ Hom π1 , SL(2, R) /SL(2, R). This has been studied
extensively and is known to be a ball of real dimension 6g − 6. It also has a structure of a complex
Banach manifold and is equipped with a Kähler metric (the well known Weil-Petersson metric) of
negative holomorphic sectional curvature.
Instead of considering representations of π1 into SL(2, R), we may consider representations to
SL(2, C). If such a representation ρ is discrete, faithful, geometrically finite and totally loxodromic
then ρ(π1 ) is quasi-Fuchsian (again in the presence of punctures purely loxodromic should be replaced with type-preserving). The collection of distinct, marked quasi-Fuchsian representations, up
to conjugation in SL(2, C) is called quasi-Fuchsian space Q = Q(Σ) ⊂ Hom π1 , SL(2, C) /SL(2, C).
A quasi-Fuchsian representation corresponds to a three dimensional hyperbolic structure on an interval bundle over Σ. According to a celebrated theorem of Bers [2], Q may be identified with the
product of two copies of Teichmüller space, and so has dimension 12g − 12. Furthermore, Q has a
1991 Mathematics Subject Classification. 32G05, 32M05.
IDP was supported by a Marie Curie Intra-European fellowship (Contract No.MEIF-CT-2003-500074) within the
6th Community Framework Programme.
1
OPEN SETS OF MAXIMAL DIMENSION IN COMPLEX HYPERBOLIC QUASI-FUCHSIAN SPACE
2
rich geometrical and analytic structure. It is a complex manifold of dimension 6g − 6 and it is endowed with a hyper-Kähler metric whose induced complex symplectic form is the complexification
of the Weil-Petersson metric on T .
Motivated by these two examples, one may consider representations of π1 into SU(2, 1) up to
conjugation, that is Hom π1 , SU(2, 1) /SU(2, 1). A representation in Hom π1 , SU(2, 1) /SU(2, 1)
is said to be complex hyperbolic quasi-Fuchsian if it is discrete, faithful, geometrically finite and
totally loxodromic (for surfaces with punctures the last condition should be type-preserving, see
[17]). The group SU(2, 1) is a triple cover of the holomorphic isometry group of complex hyperbolic
space H2C . Thus such a representation corresponds to a complex hyperbolic structure on a disc
bundle over Σ.
We remark that if ρ : π1 −→ SU(2, 1) is totally loxodromic and ρ(π1 ) neither fixes a point of
∂H2C nor preserves a totally geodesic subspace of H2C , then ρ(π1 ) is automatically discrete, see
Corollary 4.5.2 of [4]. This constrasts with the case of representations to SL(2, C). In our definition
of complex hyperbolic quasi-Fuchsian we have included the conditions that such a representation
should be both discrete and totally loxodromic. We have chosen to do so both for clarity and to
emphasise the similarity with the classical case of quasi-Fuchsian representations in SL(2, C). In
our proof we verify discreteness directly.
Bowditch has discussed notions of geometrical finiteness for variable negative curvature in [3].
In particular, if Γ is a discrete subgroup of SU(2, 1) and Ω ⊂ ∂H2C is the domain of discontinuity of
Γ then consider the orbifold MC (Γ) = H2C ∪ Ω /Γ. Bowditch defines Γ to have property F1, that
is Γ is geometrically finite in the first sense, if MC (Γ) has only finitely many topological ends, each
of which is a parabolic end. In our context, Γ will be totally loxodromic and so will have property
F1 provided MC (Γ) is a closed manifold.
The space of all marked complex hyperbolic quasi-Fuchsian representations, up to conjugacy, will
be called complex hyperbolic quasi-Fuchsian space QC = QC (Σ) ⊂ Hom π1 , SU(2, 1) /SU(2, 1).
Compared to Teichmüller space and quasi-Fuchsian space, relatively little is known about complex
hyperbolic quasi-Fuchsian space QC .
There are two ways to make a Fuchsian representation act on H2C . These correspond to the
two types of totally geodesic, isometric embeddings of the hyperbolic plane into H2C . Namely,
totally real Lagrangian planes, which may be thought of as copies of H2R , and complex lines,
which may be thought of as copies of H1C . If a discrete, faithful representation ρ is conjugate to a representation ρ : π1 −→ SO(2, 1) < SU(2, 1) then it preserves a Lagrangian plane
and is called R-Fuchsian. If a discrete, faithful representation ρ is conjugate to a representation
ρ : π1 −→ S U(1) × U(1, 1) < SU(2, 1) then it preserves a complex line and is called C-Fuchsian.
There is an important invariant of a representation ρ : π1 −→ SU(2, 1) called the Toledo invariant
denoted τ (ρ). The main properties of the Toledo invariant are
(i)
(ii)
(iii)
(iv)
(v)
τ varies continuously with ρ,
2 − 2g ≤ τ (ρ) ≤ 2g − 2, see [6],
τ (ρ) ∈ 2Z, see [15],
ρ is C-Fuchsian if and only if |τ (ρ)| = 2g − 2, see [20],
if ρ is R-Fuchsian then τ (ρ) = 0, see [15].
Further properties of complex hyperbolic representations of surface groups which refer to the Toledo
invariant are
(vi) for each even integer t with 2−2g ≤ t ≤ 2g −2 there exists a discrete, faithful representation
ρ of π1 with τ (ρ) = t, see [15],
OPEN SETS OF MAXIMAL DIMENSION IN COMPLEX HYPERBOLIC QUASI-FUCHSIAN SPACE
3
(vii) if τ (ρ1 ) = τ (ρ2 ) then ρ1 and ρ2 lie in the same component of Hom π1 , SU(2, 1) /SU(2, 1),
see [22].
We remark that in the case where Σ has cusps then, in fact, τ (ρ) is a real number in the interval
χ(Σ), −χ(Σ) and for any real number t in this interval there exists a discrete, faithful representation ρ of π1 (Σ) with τ (ρ) = t, see [17]. Moreover, Dutenhefner and Gusevskii [7] have constructed
an example of a discrete, faithful, type-preserving representation of the fundamental group of a
particular punctured surface whose limit set is a wild knot. This means that it cannot be in the
same component of the space of discrete faithful representations as a Fuchsian representation. It
may well be possible to extend this example to the case of closed surfaces, which would lead to questions about the number of components of complex hyperbolic quasi-Fuchsian space (Xia’s result
[22], given in (vii) above, does not involve discreteness).
An immediate consequence of (i) and (iii) is that τ is locally constant and, together with (iv),
implies that given a C-Fuchsian representation ρ0 any nearby representation ρt is also C-Fuchsian.
This result is known as the Toledo-Goldman rigidity theorem [20], [13]. In fact, the component of
Hom π1 , SU(2, 1) /SU(2, 1) with |τ | = 2g − 2 has dimension 8g − 6 and the other components have
dimension 16g − 16 (see Theorem 6 of [13]).
In this paper we begin with any R-Fuchsian representation ρ0 and we consider nearby represen
tations ρt in Hom π1 , SU(2, 1) /SU(2, 1). Our main result is:
Theorem 1.1. Let Σ be a closed surface of genus g with fundamental group π1 = π1 (Σ). Let
ρ0 : π1 −→ SU(2, 1) be an R-Fuchsian representation of π1 . Then there exists an open neighbourhood
U = U (ρ0 ) of ρ0 in Hom π1 , SU(2, 1) /SU(2, 1) so that any representation ρt in U is complex
hyperbolic quasi-Fuchsian (that is discrete, faithful, geometrically finite and totally loxodromic).
This theorem may be thought of as an instance of structural stability, see Sullivan [19]. However,
it is not clear how to generalise the details of Sullivan’s method from subgroups of SL(2, C) to
subgroups of SU(2, 1). Therefore we use a different method.
An immediate consequence of Theorem 1.1 is:
Corollary 1.2. There are open sets of dimension 16g − 16 in QC (Σ).
Up to now, families of complex hyperbolic quasi-Fuchsian groups have only been constructed
by varying a particular geometrical construction, see for example [16], [17], [9], [10], [11], [18].
By contrast, in this paper we only use the hypothesis that ρt and ρ0 are nearby representations.
From this information we must make a geometrical construction of a fundamental domain. To go
from algebra to geometry (and back again) we use the theorem of Falbel and Zocca [12], Theorem
2.1. We prove Theorem 1.1 by first constructing a fundamental domain ∆0 in H2C for ρ0 (π1 )
and then showing that for any other representation ρt sufficiently close to ρ0 we may construct a
fundamental domain ∆t for ρt (π1 ). By sufficiently close, we mean that there exists an > 0 so
that the generators of ρt (π1 ) are -close to the generators of ρ0 (π1 ) in the l2 -topology on SU(2, 1).
Constructing fundamental domains in complex hyperbolic space is challenging because, unlike
the case of constant curvature, there are no totally geodesic real hypersurfaces. Thus, before
constructing a fundamental polyhedron we must choose the class of real hypersurfaces containing
its faces. The most usual method of constructing a fundamental domain in complex hyperbolic
space involves domains whose boundary is made up of pieces of bisectors. In particular, this is the
case for the construction of Dirichlet domains. This idea goes back to Giraud and was developed
further by Mostow and Goldman (see [14] and the references therein), and see [16], [17] for other
examples of fundamental domains bounded by bisectors. Other classes of hypersurfaces used to
build fundamental domains are C-spheres [12] and R-spheres [18] (for the relationship between
C-spheres and R-spheres see [11]).
OPEN SETS OF MAXIMAL DIMENSION IN COMPLEX HYPERBOLIC QUASI-FUCHSIAN SPACE
4
Since bisectors are rather badly adapted to R-Fuchsian representations, we have chosen to introduce a new class of hypersurfaces. Just as bisectors are foliated by slices that are complex lines
so our hypersurfaces are foliated by Lagrangian planes. These hypersurfaces resemble a pack of
(infinitely many) playing cards, each Lagrangian plane representing a card. Therefore we call we
call such hypersurfaces packs. The boundaries of packs are foliated by R-circles and so are closely
related to Schwartz’ R-spheres [18] and examples of packs (with no twist) were introduced by Will
[21], who calls them R-balls. Both Schwartz and Will use these objects to construct fundamental
domains. The relationship between bisectors and packs is an example of the duality, which resembles mirror symmetry, between complex and real objects in complex hyperbolic space, see the
discussion in the introduction to [11]. The polyhedra ∆0 and ∆t we construct have boundaries that
are made up of pieces of packs. In order to show that ρt (π1 ) is complex hyperbolic quasi-Fuchsian
we give a version of Poincaré’s polyhedron theorem, Theorem 4.2, for such polyhedra (this should
be compared with [8]).
2. Preliminaries
2.1. Complex Hyperbolic Space. Let C2,1 be the vector space C3 with the Hermitian form of
signature (2, 1) given by
hz, wi = w∗ Jz = z1 w1 + z2 w2 − z3 w3 .
Its matrix is


1 0 0
J =  0 1 0 .
0 0 −1
Consider the following subspaces of C2,1 :
n
o
V− =
z ∈ C2,1 : hz, zi < 0 ,
n
o
V0 =
z ∈ C2,1 − {0} : hz, zi = 0 .
Let P : C2,1 − {0} −→ CP 2 be the canonical projection onto complex projective space. Then
complex hyperbolic space H2C is defined to be PV− and its boundary ∂H2C is PV0 . Specifically,
C2,1 − {0} may be covered with three charts H1 , H2 , H3 where Hj comprises those points in
C2,1 − {0} for which zj 6= 0. It is clear that V− and V0 are both contained in H3 . The canonical
projection from H3 to C2 is given by P(z) = (z1 /z3 , z2 /z3 ). Therefore we can write H2C = P(V− )
and ∂H2C = P(V0 ) as
H2C = (z1 , z2 ) ∈ C2 : |z1 |2 + |z2 |2 < 1 ,
∂H2C = (z1 , z2 ) ∈ C2 : |z1 |2 + |z2 |2 = 1 .
In other words, H2C is the unit ball in C2 and likewise ∂H2C is the unit sphere S 3 .
Conversely, given a point z of C2 = P(H3 ) ⊂ CP 2 we may lift z = (z1 , z2 ) to a point z in
H3 ⊂ C2,1 , called the standard lift of z, by writing z in non-homogeneous coordinates as
 
z1
z = z2  .
1
The Bergman metric on H2C is defined by the distance function ρ given by the formula
hz, wi
ρ(z,
w)
hz,
wi
hw,
zi
cosh2
=
= 2 2
2
hz, zi hw, wi
|z| |w|
OPEN SETS OF MAXIMAL DIMENSION IN COMPLEX HYPERBOLIC QUASI-FUCHSIAN SPACE
where z and w in V− are the standard lifts of z and w in H2C and |z| =
4
hz, zi hdz, zi
2
ds = −
det
.
hz, dzi hdz, dzi
hz, zi2
p
5
−hz, zi. Alternatively,
The holomorphic sectional curvature of H2C equals to −1 and its real sectional curvature is pinched
between −1 and −1/4.
There are no totally geodesic, real hypersurfaces of H2C , but there are are two kinds of totally
geodesic 2-dimensional subspaces of complex hyperbolic space, (see Section 3.1.11 of [14]). Namely:
(i) complex lines L, which have constant curvature −1, and
(ii) totally real Lagrangian planes R, which have constant curvature −1/4.
Both of these subspaces are isometrically embedded copies of the hyperbolic plane.
2.2. Isometries. Let U(2, 1) be the group of unitary matrices for the Hermitian form h·, ·i. Each
T
such matrix A satisfies the relation A−1 = JA∗ J where A∗ = A .
The full group of holomorphic isometries of complex hyperbolic space is the projective unitary
group PU(2, 1) = U(2, 1)/U(1), where U(1) = {eiθ I, θ ∈ [0, 2π)} and I is the 3 × 3 identity matrix.
For our purposes we shall consider instead the group SU(2, 1) of matrices which are unitary with
respect to h·, ·i, and have determinant 1. Therefore PU(2, 1) = SU(2, 1)/{I, ωI, ω 2 I}, where ω is a
non real cube root of unity, and so SU(2, 1) is a 3-fold covering of PU(2, 1).
Every complex line L is the image under some A ∈ SU(2, 1) of the complex line where the first
coordinate is zero. The subgroup of SU(2, 1) stabilising this particular complex line is thus the
group of block diagonal matrices S U(1) × U(1, 1) < SU(2, 1). Similarly, every Lagrangian plane
is the image under some element of SU(2, 1) of the Lagrangian plane RR where both coordinates
are real, called the standard real Lagrangian plane. This is preserved by the subgroup of SU(2, 1)
comprising matrices with real entries, that is SO(2, 1) < SU(2, 1).
Holomorphic isometries of H2C are classified as follows.
(i) An isometry is loxodromic if it fixes exactly two points of ∂H2C .
(ii) An isometry is parabolic if it fixes exactly one point of ∂H2C .
(iii) An isometry is elliptic if it fixes at least one point of H2C .
The complex conjugation map ιR : (z1 , z2 ) 7−→ (z 1 , z 2 ) is an involution of H2C fixing the standard
real Lagrangian plane RR . It too is an isometry. Indeed any anti-holomorphic isometry of H2C may
be written as ιR followed by some element of PU(2, 1). Any Lagrangian plane may be written as
R = B(RR ) for some B ∈ SU(2, 1) and so ι = BιR B −1 is an anti-holomorphic isometry of H2C fixing
R.
Falbel and Zocca [12] have used involutions fixing Lagrangian planes to give the following characterisation of elements of SU(2, 1):
Theorem 2.1. Any element C of SU(2, 1) may be written as C = ι1 ◦ ι0 where ι0 and ι1 are
involutions fixing Lagrangian planes R0 and R1 respectively. Moreover
(i) C = ι1 ◦ ι0 is loxodromic if and only if R0 and R1 are disjoint;
(ii) C = ι1 ◦ ι0 is parabolic if and only if R0 and R1 intersect in exactly one point of ∂H2C ;
(iii) C = ι1 ◦ ι0 is elliptic if and only if R0 and R1 intersect in at least one point of H2C .
We conclude this section by considering the case where C is loxodromic in more detail. Since
elements of SU(2, 1) preserve the Hermitian form, it is not hard to show that if µ is an eigenvalue
of A ∈ SU(2, 1) then so is µ−1 (Lemma 6.2.5 of [14]). From this fact we find that A is loxodromic
if and only if one of its eigenvalues µ satisfies |µ| > 1. In particular, if |tr(A)| > 3 then A is
OPEN SETS OF MAXIMAL DIMENSION IN COMPLEX HYPERBOLIC QUASI-FUCHSIAN SPACE
6
loxodromic. We will use this fact repeatedly. Goldman gives a more precise statement in Theorem
6.2.4 of [14], but we will not need this level of detail.
If C ∈ SU(2, 1) is loxodromic then one of its eigenvalues is µ = eδ−iφ where δ > 0 and φ ∈ (−π, π].
Hence another eigenvalue of C is µ−1 = e−δ−iφ . Since det(C) = 1, its third eigenvalue must be
e2iφ . Therefore tr(C) = 2 cosh(δ)e−iφ + e2iφ . The eigenvectors corresponding to µ and µ−1 span a
complex line L in H2C . This line is called the complex axis of C and is written L = LC = Ax(C).
In fact we may write


cosh(δ)e−iφ
0
sinh(δ)e−iφ
 Q−1
(2.1)
C = Q
0
e2iφ
0
sinh(δ)e−iφ
0 cosh(δ)e−iφ
for some Q ∈ SU(2, 1). If C lies in SO(2, 1) and corresponds to a a loxodromic isometry of the
hyperbolic plane then φ = 0 and so tr(C) = 2 cosh(δ) + 1 is real and greater than 3. (If φ = π then
C corresponds to a hyperbolic glide reflection on H2R and tr(C) = −2 cosh(δ) + 1 < −1.)
Lemma 2.2. Suppose that C ∈ SU(2, 1) is loxodromic with eigenvalues eδ−iφ , e−δ−iφ , e2iφ then C
may be written as
C = e2iφ I + sinh(δ)e−iφ E + cosh(δ)e−iφ − e2iφ E 2
for some matrix E satisfying E 3 = E and JE ∗ J = −E.
Proof. This immediately follows from (2.1) writing


0 0 1
E = Q 0 0 0 Q−1 .
1 0 0
The group SU(2, 1) is a topological space equipped with the compact-open topology. This is
equivalent to the l2 -topology on SU(2, 1) ⊂ C9 and thus SU(2, 1) is a Hilbert space with inner
product given by
hhA, Bii = < tr(AB ∗ )
for every A, B ∈ SU(2, 1). Let k · k denote the respective l2 -norm. We note that for every
A ∈ SU(2, 1) we have kAk = kA−1 k.
The Lie algebra su(2, 1) of the complex Lie group SU(2, 1) consists of matrices D satisfying the
relations JD∗ J = −D and tr(D) = 0. Actually, every element of su(2, 1) is a zero trace matrix of
the form
0 ∗ D z
D=
,
z iθ
where D0 ∈ u(2), θ ∈ R, z is a vector in C2 and z ∗ is its Hermitian transpose, see page 103 of [14].
In what follows, su(2, 1) will also be considered as a Hilbert space equipped with the l2 -norm. The
mapping
∞
X
Dn
exp(D) =
n!
n=0
takes D ∈ su(2, 1) to SU(2, 1) and is called the exponential mapping.
Lemma 2.3. Suppose that C and E are as given in Lemma 2.2. Then C = exp(D) where
D = 2iφI + δE − 3iφE 2 . Moreover, E ∈ su(2, 1).
OPEN SETS OF MAXIMAL DIMENSION IN COMPLEX HYPERBOLIC QUASI-FUCHSIAN SPACE
7
Proof. From the proof of Lemma 2.2 it is clear that E and 2iI − 3iE 2 are in su(2, 1). Using E 3 = E
and expanding we obtain
exp(D) = exp(2iφI + δE − 3iφE 2 )
= exp(δE) exp(2iφI − 3iφE 2 )
2 2iφ
−iφ 2
2iφ 2
=
I + sinh(δ)E + cosh(δ) − 1 E
e I +e E −e E
= e2iφ I + sinh(δ)e−iφ E + cosh(δ)e−iφ − e2iφ E 2 .
2.3. Projection onto Lagrangian planes. In this section we consider totally real Lagrangian
planes R. We discuss orthogonal projection ΠR onto R and its fibres Π−1 (z). First, following
Goldman, we give a formula for the midpoint of two points of complex hyperbolic space.
Proposition 2.4. Let z, w be any points of V− ⊂ C2,1 and z = Pz, w = Pw be the corresponding
points of H2C . Let
(2.2)
m=
hz, wi
1
w.
z − |z|
hz, wi |w|
Then m ∈ V− and, writing m = Pm, we have ρ(m, z) = ρ(m, w) = ρ(z, w)/2.
If m is as defined in Proposition 2.4 then we call m the midpoint of z and w (see Exercise 3.1.4
of [14]).
Proof. First, observe that
2hz, wi
hm, mi = −2 −
= −2 1 + cosh ρ(z, w)/2 = −4 cosh2 ρ(z, w)/4 .
|z| |w|
Thus m ∈ V− and m = Pm ∈ H2C and we write |m| =
hm, zi =
p
−hm, mi = 2 cosh ρ(z, w)/4 . Moreover,
hz, zi hz, wihw, zi
−
|z|
|hz, wi| |w|
= −|z| −
|hz, wi|
|w|
= −|z| 1 + cosh ρ(z, w)/2
= −|z| 2 cosh2 ρ(z, w)/4
= −|z| |m| cosh ρ(z, w)/4 .
Therefore
hm, zi
cosh ρ(m, z)/2 =
= cosh ρ(z, w)/4 .
|z| |m|
OPEN SETS OF MAXIMAL DIMENSION IN COMPLEX HYPERBOLIC QUASI-FUCHSIAN SPACE
8
Similarly
hm, wi =
=
=
hz, wi hz, wihw, wi
−
|z|
|hz, wi| |w|
|hz, wi|
hz, wi
|w| +
|hz, wi|
|z|
hz, wi
|w| |m| cosh ρ(z, w)/4
|hz, wi|
and so
hm, wi
= cosh ρ(z, w)/4 .
cosh ρ(m, w)/2 =
|w| |m|
Hence ρ(m, z) = ρ(m, w) = ρ(z, w)/2 as required.
We use Proposition 2.4 to derive a formula for the orthogonal projection onto a Lagrangian
plane R (see Section 3.3.6 of [14]). Let ιR denote the (anti-holomorphic) reflection in R. Then the
orthogonal projection ΠR (z) of any z ∈ H2C onto R is defined to be the midpoint m of the points
z and ιR (z). That is, if z ∈ V− is a lift of z then
hz, ιR (z)i
1
z−
ιR (z) .
ΠR (z) = P
|z|
|hz, ιR (z)i| |ιR (z)|
Proposition 2.5. Let R be a Lagrangian plane stabilised by the subgroup GR of SU(2, 1). Then,
for every A ∈ GR
A ◦ ΠR = ΠR ◦ A.
Consequently, if w ∈ R,
−1
Π−1
R A(z) = A ΠR (z) .
Proof. Let z ∈ H2C . Then, ΠR (z) = m is the midpoint of z and ι(z). Hence
ρ A(z), A(m) = ρ(z, m) = ρ ι(z), m = ρ Aι(z), A(m) .
Also
ρ A(z), Aι(z) = ρ z, ι(z) = 2ρ(m, z) = 2ρ A(m), A(z) .
Thus A(m) is the midpoint of A(z) and Aι(z). But since Aι(z) = ιA(z) we see that
ΠR A(z) = A(m) = A ΠR (z) .
Now suppose that w ∈ R and choose any z with ΠR (z) = w. Then
A(w) = AΠR (z) = ΠR A(z).
−1
Thus A(z) ∈ ΠR A(w) and so AΠR −1 (w) ⊂ ΠR −1 A(w). Similarly if z 0 is chosen so that
ΠR (z 0 ) = A(w) then
w = A−1 ΠR (z 0 ) = ΠR A−1 (z 0 )
and so z 0 ∈ AΠR −1 (w). Hence ΠR −1 A(w) ⊂ AΠR −1 (w).
We consider the special case where R is the standard real Lagrangian plane RR , that is
RR = H2R = (z1 , z2 ) ∈ H2C : =(z1 ) = =(z2 ) = 0
and we denote orthogonal projection onto RR by ΠR . Consider a point z = (z1 , z2 ) ∈ H2C . Then
reflection ιR in RR is given by
ιR (z) = z = z 1 , z 2 .
OPEN SETS OF MAXIMAL DIMENSION IN COMPLEX HYPERBOLIC QUASI-FUCHSIAN SPACE
9
Following Goldman (page 108 of [14]), we write
η(z)2 = −hz, ιR zi = 1 − z1 2 − z2 2 .
Observe that 0 < 1 − |z1 |2 − |z2 |2 ≤ < 1 − z1 2 − z2 2 = < η(z)2 , and in particular, η(z)2 6= 0.
Applying (2.2) we find that the midpoint m = (m1 , m2 ) of z and ιR (z) is given by
2 + η(z)2
η(z)
<
z
η(z)2 zk + η(z)2 z k
k
,
= 2η(z)2 mk = 2
2
η(z) + η(z)
η(z)2 + η(z)2 for k = 1, 2. Clearly, m lies on RR , and if z ∈ RR , then ΠR (z) = z.
Corollary 2.6. ΠR is real analytic.
The subgroup of SU(2, 1) stabilising RR comprises those matrices with all real entries, that is
SO(2, 1) the isometry group of the hyperbolic plane. Proposition 2.5 immediately implies that ΠR
commutes with all elements of SO(2, 1).
Proposition 2.7. If RR is the standard real Lagrangian plane
RR = (z1 , z2 ) ∈ H2C : =(z1 ) = =(z2 ) = 0 ,
then ΠR −1 (0, 0) is the purely imaginary Lagrangian plane
RJ = (z1 , z2 ) ∈ H2C : <(z1 ) = <(z2 ) = 0 .
Proof. If z1 and z2 are both purely imaginary then η(z)2 = 1 − z12 − z22 is a positive real number.
It is clear from the above construction that
m1 = <(z 1 ) = 0,
m2 = <(z 2 ) = 0.
Thus the Lagrangian plane RJ is contained in ΠR −1 (0, 0).
2
Conversely, the set Π−1
R (0, 0) is the collection of points (z1 , z2 ) ∈ HC satisfying
η(z)2 z1 + η(z)2 z 1 = η(z)2 z2 + η(z)2 z 2 = 0.
When z1 and z2 are both non-zero, these two equations are equivalent to
z1 2
z2 2
−η(z)2
=
=
.
η(z)2
|z1 |2
|z2 |2
Writing z1 2 = |z1 |2 eiφ and z2 2 = |z2 |2 eiφ we obtain
η(z)2 = −η(z)2 e−iφ = − 1 − z1 2 − z2 2 )e−iφ = −e−iφ + |z1 |2 + |z2 |2 .
Therefore eiφ ∈ R. Since |z1 |2 + |z2 |2 < 1 we see that eiφ = −1. Thus z1 and z2 are both purely
imaginary. When one of z1 or z2 is zero, a similar argument shows that the other one is purely
imaginary (or zero). Thus ΠR −1 (0, 0) is contained in the Lagrangian plane RJ .
Using the fact that SU(2, 1) acts transitively on the set of Lagrangian planes in H2C we immediately have:
Corollary 2.8. Let w be any point on the Lagrangian plane R. Then ΠR −1 (w) is a Lagrangian
plane.
Corollary 2.9. For every Lagrangian plane R, the orthogonal projection ΠR is real analytic.
OPEN SETS OF MAXIMAL DIMENSION IN COMPLEX HYPERBOLIC QUASI-FUCHSIAN SPACE
10
3. Packs
In this section we introduce real analytic 3-(real) dimensional submanifolds of complex hyperbolic
space which are foliated by Lagrangian planes. These submanifolds can be considered as the
counterparts of bisectors. (For an extensive treatment of the latter, see [14]).
Let C be a loxodromic map in SU(2, 1) given, as in Lemma 2.2, by
C = e2iφ I + sinh(δ)e−iφ E + cosh(δ)e−iφ − e2iφ E 2 = exp(D)
where E ∈ su(2, 1) satisfies E 3 = E. For any x ∈ R define C x by
C x = e2ixφ I + sinh(xδ)e−xiφ E + cosh(xδ)e−xiφ − e2ixφ E 2 = exp(xD).
Observe that C x has the same eigenvectors as C, but its eigenvalues are the eigenvalues of C raised
to the xth power. Hence we immediately see that C x is a loxodromic element of SU(2, 1) for all
x ∈ R − {0} and C 0 = I. Moreover, for any integer n it is clear that C n agrees with the usual
notion of the nth power of C. This justifies the use of a superscript.
Proposition 3.1. Let R0 and R1 be disjoint Lagrangian planes in H2C and let ι0 and ι1 be the
respective inversions. Consider C = ι1 ι0 (which is loxodromic map by Theorem 2.1) and its powers
C x for each x ∈ R. Then:
(i)
(ii)
(iii)
(iv)
ιx defined by C x = ιx ι0 is inversion in a Lagrangian plane Rx = C x/2 (R0 ).
Rx intersects the complex axis LC of C orthogonally in a geodesic γx .
The geodesics γx are the leaves of a foliation of LC .
For each x 6= y ∈ R, Rx and Ry are disjoint.
Proof. Since ι0 Cι0 = C −1 we also have ι0 C x ι0 = C −x . Thus ιx = C x ι0 has order 2 and so is
involution in a Lagrangian plane. Then inversion in C x/2 (R0 ) is
C x/2 ι0 C −x/2 = (ιx/2 ι0 )ι0 (ι0 ιx/2 ) = ιx/2 ι0 ιx/2 = (ιx/2 ι0 )2 ι0 = (C x/2 )2 ι0 = C x ι0 = ιx .
Part (i) follows by construction. Likewise, parts (ii) and (iii) follow immediately. Finally, ιx ι0 = C x
and ιy ι0 = C y and so ιx ιy = C x C −y = C x−y which is loxodromic, proving (iv).
Definition 3.2. Given disjoint Lagrangian planes R0 and R1 , then for each x ∈ R let Rx be the
Lagrangian plane constructed in Proposition 3.1. Define
[
P = P (R0 , R1 ) =
Rx .
x∈R
Then P is a real analytic 3-submanifold which we call the pack determined by R0 and R1 . We call
γ = Ax(ι1 ι0 ) the spine of P and the Lagrangian planes Rx for x ∈ R the slices of P .
Observe that P contains L, the complex line containing γ, the spine of P . We remark that
packs are analogous to bisectors, but with Lagrangian planes for slices rather than complex lines.
The following proposition is obvious from the construction and emphasises the similarity between
bisectors and packs (compare it with Section 5.1.2 of [14]). The definition of packs associated to
loxodromic maps C that preserve a Lagrangian plane (that is with φ = 0) was given by Will [21].
Proposition 3.3. Let P be a pack. Then P is homeomorphic to a 3-ball whose boundary lies in
∂H2C . Moreover, H2C − P , the complement of P , has two components, each homeomorphic to a
4-ball.
OPEN SETS OF MAXIMAL DIMENSION IN COMPLEX HYPERBOLIC QUASI-FUCHSIAN SPACE
11
We remark that the boundary of P contains the boundary of the complex line L and is foliated
by the boundaries of the Lagrangian planes Rx . Since it is also homeomorphic to a sphere, it is an
example of an R-sphere (hybrid sphere), see [11, 18].
Let L be a complex line and ΠL be orthogonal projection onto L. Let γ be a geodesic contained
in L. Then, following Mostow, the bisector with spine γ is the inverse image of γ under ΠL .
Moreover, each slice of this bisector is the inverse image of a point of γ under ΠL , and is a complex
line. Following Will, Section 6.1.1 of [21], we now show that performing the same construction but
for a Lagrangian plane R gives a pack.
Proposition 3.4. Suppose that the geodesic γ lies on a totally real plane R. Then the set
[
P (γ) = Π−1
Π−1
R (γ) =
R (z).
z∈γ
−1
is the pack determined by the Lagrangian planes R0 = Π−1
R (z0 ) and R1 = ΠR (z1 ) for any distinct
points z0 , z1 ∈ γ. Moreover, for each z ∈ γ, the Lagrangian plane Π−1
R (z) is a slice of P (γ).
−1
Proof. Choose any points z0 and z1 on γ and let R0 = Π−1
R (z0 ) and R1 = ΠR . The involutions
ι0 and ι1 fixing R0 and R1 preserve R. Hence C = ι1 ι0 is a loxodromic map with real trace and
commuting with ΠR . Since the axis of C is γ, any point on z on γ has the form z = C x (z0 ) for
some x ∈ R. The result follows.
4. Poincaré’s Theorem
Definition 4.1. Let Γ be a discrete group of complex hyperbolic isometries. A subset ∆ of H2C is
called a fundamental domain for Γ if the following hold.
(i)
(ii)
(iii)
(iv)
2
∆ is a domain in HC , that is an open connected set;
∆ ∩ A(∆) = ∅ for all A ∈ Γ \ {I};
S
2
A∈Γ A(∆) = HC ;
the complex hyperbolic volume of ∂∆ is 0.
In this section we establish a Poincaré’s theorem suitable for our purposes, compare [8]. Let P
be a pack then the complement of P consists of two half-spaces. We consider polyhedra ∆ obtained
by intersecting finitely many such half-spaces.
A natural cell decomposition exists for the closure of ∆ given by the intersections of the defining
packs. The cells of this decomposition are called the faces of ∆. The codimension 1 faces are called
the sides and will be denoted S. The codimension 2 faces are called the edges of ∆ and will be
denoted R (because in the applications each edge R will be a Lagrangian plane).
We also consider the action of Γ on ∂H2C . To each codimension k face of ∆ whose closure
meets ∂H2C , we associate a codimension k subset of ∂H2C called the ideal boundary of the face. To
construct the ideal boundary, take the closure of a face and then remove the union of the closures
of all lower dimensional faces. The ideal boundary is the intersection of what remains with ∂H2C .
For example, if an edge R is a Lagrangian plane then its ideal boundary is its boundary in ∂H2C ,
which is an R-circle.
Given such a polyhedron ∆, we wish to establish conditions so that the group Γ generated by
the identifications of the sides of ∆, is discrete and that ∆ is a fundamental domain for Γ in H2C .
4.1. Side conditions. The sides of ∆ are paired by elements of SU(2, 1): for each side S of ∆,
there is a side S 0 (not necessarily distinct from S) and an element AS ∈ SU(2, 1) such that:
(S.1) AS (S) = S 0 ,
OPEN SETS OF MAXIMAL DIMENSION IN COMPLEX HYPERBOLIC QUASI-FUCHSIAN SPACE
12
(S.2) AS 0 = A−1
S ,
(S.3) AS (∆) ∩ ∆ = ∅,
(S.4) AS (∆) ∩ ∆ = S 0 .
The isometries AS are called the side pairing transformations of ∆. Let Γ be the group generated
by these transformations. We allow S and S 0 to be the same side, that is a side may be mapped
to itself. In this case condition (S.2) requires us to impose AS 2 = 1, which is called a reflection
relation. (This will not arise in our applications.)
We require that ∆ is defined by intersecting finitely many half spaces determined by non-tangent
packs. This gives two more side conditions:
(S.5) The polyhedron ∆ has only finitely many sides S, each side has only finitely many edges.
(S.6) There exists δ > 0 so that each pair of disjoint sides is at least a distance δ apart.
4.2. Edge conditions.
(E.1) Each edge of ∆ is a complete submanifold of H2C homeomorphic to an open ball of real
dimension 2.
Start with an edge R1 which lies on the boundary of two sides, call one of them S1 . Then there
is a side S10 , and a side pairing transformation A1 , with A1 (S1 ) = S10 .
Set R2 = A1 (R1 ). Like R1 , the edge R2 lies on the boundary of exactly two sides, one of them
is S10 , call the other S2 . Again, there is a side S20 , and a side pairing transformation A2 , with
A2 (S2 ) = S20 . Following this process gives rise to a sequence Rj of edges, a sequence Aj of side
pairing transformations, and a sequence (Sj , Sj0 ) of pairs of sides.
Since ∆ has a finite number of sides, the sequence of edges has to be periodic and hence all
three sequences are periodic. Let k be the least period so that all three sequences are periodic with
period k. The cyclically ordered sequence of edges R1 , . . . , Rk , is called a cycle of edges; k is the
period of the cycle. Observe that
Ak ◦ · · · ◦ A1 (R1 ) = R1 .
The element B = Ak ◦ · · · ◦ A1 is called the cycle transformation at the edge R1 .
Given a cycle transformation B as above and a positive integer m, define a sequence of mk
elements B0 , B1 , . . . , Bmk , of Γ as follows:
B0 = 1,
Bk = B,
..
.
B1 = A1 ,
Bk+1 = A1 ◦ B,
..
.
. . . Bk−1 = Ak−1 ◦ · · · ◦ A1 ,
. . . B2k−1 = Ak−1 ◦ · · · ◦ A1 ◦ B,
..
.
Bmk−k = B m−1 , Bmk−k+1 = A1 ◦ B m−1 , . . . Bmk−1 = Ak−1 ◦ · · · ◦ A1 ◦ B m−1 .
We define F(B) to be the following family of polyhedra:
−1
F(B) = ∆, B1−1 (∆), . . . , Bmk−1
(∆) .
(E.2) For each edge R, the restriction to R of the cycle element B = BR at R is the identity,
and there is a positive integer m so that B m = 1, that is, B has order m. Moreover, the
polyhedra of the family F(B) fit together without overlap, and their closures fill out a closed
neighbourhood of the edge R.
The relations in Γ of the form B m = 1, are called the cycle relations.
We now give the main theorem of this section. See also Theorem 3.2 of [17] for a similar theorem
for polyhedra whose faces are contained in bisectors (and which may have tangencies between the
faces).
Theorem 4.2. Assume that the finite sided polyhedron ∆ ⊂ H2C with side pairing transformations
Ai satisfies all conditions (S.1) to (S.6), (E.1) and (E.2). Then:
OPEN SETS OF MAXIMAL DIMENSION IN COMPLEX HYPERBOLIC QUASI-FUCHSIAN SPACE
(i)
(ii)
(iii)
(iv)
(v)
13
the group Γ generated by these transformations is discrete;
∆ is a fundamental domain for Γ;
the reflection relations and cycle relations form a complete set of relations for Γ;
Γ contains no parabolic elements;
Γ is geometrically finite.
The proof of (i), (ii) and (iii) follows that in Epstein and Petronio, Theorem 4.14 [8]. Observe
that, using Remark 3.24 of [8] our conditions (S.5) and (S.6) imply the condition (Metric) of Epstein
and Petronio. If Γ had a parabolic element then, necessarily its fixed point would lie in the closure
of ∆, compare Theorem 10.3.2 of [1], and this fixed point would lie on the ideal boundary of (at
least) two disjoint faces of ∆. This contradicts (S.6). It is clear that the the ideal boundary of ∆
is contained in the region of discontinuity Ω(Γ) for the action of Γ on ∂H2C . Also, using (S.4) each
point on the ideal boundary of a side S has a neighbourhood covered by the ideal boundaries of
S, ∆ and AS −1 (∆). Hence it too is contained in Ω(Γ). Finally, using (E.2) a similar argument
shows that each point in the ideal boundary of an edge R is also in Ω(Γ). Therefore (H2C ∪ Ω)/Γ is
a closed manifold. Using definition F1 of Bowditch [3] (see the discussion in the introduction) we
see that Γ is geometrically finite.
Corollary 4.3. Suppose that the group Γ from Theorem 4.2 is a representation of π1 , the fundamental group of a surface of genus g. Suppose that the cycle relations and reflection relations in
(iii) introduce no new relations. Then Γ is complex hyperbolic quasi-Fuchsian.
We note that if π1 is the fundamental group of a punctured surface then (S.6) does not hold.
5. Proof of the main theorem
5.1. A fundamental polyhedron for an R-Fuchsian group. Let Σ be a closed surface of genus
g > 1 and let ρ0 be any R-Fuchsian representation of π1 , the fundamental group of Σ. We denote
the image of ρ0 by Γ0 = ρ0 (π1 ) < SU(2, 1). Without loss of generality, we suppose that Γ0 preserves
RR and so Γ0 < SO(2, 1). Consider the action of Γ0 on RR and let ∆0 be a fundamental hyperbolic
polygon for this action with 4g sides s(1) , . . . , s(4g) . Let v (1) , . . . , v (4g) denote the vertices of ∆0 .
We adopt the convention that s(k) has endpoints v (k) and v (k+1) and superscripts are taken mod
4g. Conjugating if necessary, we suppose that v (1) is the origin o. By construction, there are 4g
(1)
(4g)
elements of Γ0 , denoted A0 , . . . , A0
that pair the sides of ∆ according to the following rules:
(4j+1)
(i) For j = 0, . . . , g − 1 the map A0
(4j+2)
sends the side s(4j+1) to the side s(4j+3) and the
(4j+1)
(4j+3) −1
to the side s(4j+4) . Thus A0
= A0
and
sends the side s(4j+2)
(4j+2)
(4j+4) −1
A0
= A0
.
(ii) There are no reflection relations and only one cycle relation:
map A0
(5.1)
g−1
Y
(4j+2)
A0
(4j+1) −1
A0
(4j+2) −1
A0
(4j+1)
A0
= I.
j=0
For this polygon, it is straightforward to verify that side conditions analogous to (S.1) to (S.6)
are satisfied. In this case, each codimension 2 face is a point, namely one of v (1) , . . . , v (4g) . This
condition replaces (E.1). With this change, (E.2) is also satisfied. Thus we could have used the
classical Poincaré polygon theorem to verify that ∆0 is a fundamental domain for the action of Γ0
on RR . Moreover, as (5.1) generates all relations in π1 we see that ρ0 is faithful. In particular,
Γ0 has no elliptic elements. Since there are no tangencies between faces of ∆0 we also see that Γ0
contains no parabolics. Hence it is totally loxodromic.
OPEN SETS OF MAXIMAL DIMENSION IN COMPLEX HYPERBOLIC QUASI-FUCHSIAN SPACE
14
Let ∆0 = Π−1
R (∆0 ) be the inverse image of the polygon ∆0 under projection onto RR (see
Section 6.1.1 of [21] where Will constructs fundamental domains for R-Fuchsian triangle groups
and punctured torus groups in a similar way). We claim that ∆0 satisfies the conditions (S.1)
to (S.6), (E.1) and (E.2). Thus Theorem 4.2 will imply that ∆0 is a fundamental domain for the
action of Γ0 on H2C . We now show how to check the conditions. The edges of ∆0 are the Lagrangian
(k)
(1)
(k) ). In particular, R
= Π−1
0
R (v
planes R0
(k)
The sides of ∆0 are S0
= Π−1
R (o) = RJ . Thus condition (E.1) is satisfied.
(k)
(k) ) for k = 0, . . . , 4g. These are each pieces of the pack P
= Π−1
0
R (v
(k)
(k+1)
determined by the Lagrangian planes R0 and R0
.
By Proposition 2.5, ΠR commutes with any element of SO(2, 1), and so for j = 0, . . . , g − 1
(4j+1)
the map A0
(4j+2)
S0
(4j+1)
sends the side S0
(4j+3)
to the side S0
(4j+2)
and the map A0
sends the side
(4j+4)
S0
.
to the side
Thus the side conditions (S.1) to (S.6) are automatically satisfied. The
condition (E.2) is therefore satisfied: there is only one cycle of vertices and the cycle transformation
is given by (5.1) with m = 1. Using Poincaré’s theorem, Theorem 4.2, we see that ∆0 is indeed a
fundamental domain for Γ0 .
(k)
(1)
By construction, for any k = 1, . . . , 4g the edge R0 is the image of R0 = RJ under some fixed
(1)
(4g)
word in the generators A0 , . . . , A0
. In fact this word comprises the last n letters of the relation
(k)
(1)
(4)
(1)
(5.1) for some n. We denote this word by B0 . For example B0 is the identity, B0 = A0 and
(3)
(2) −1 (1)
(k)
B0 = A0
A0 . There is a homotopy class of loops βk ∈ π1 so that B0 = ρ0 (βk ). Clearly
(k)
B0
(k)
is loxodromic for each k. So there is a constant K > 0 so that tr(B0 ) ≥ 3 + K > 3 for all k.
5.2. The variation of the polyhedron. Let Γt = ρt (π1 ) < SU(2, 1) be a point in the represen
tation variety Hom (π1 , SU(2, 1) /SU(2, 1). We will only consider representations that are close
(k)
to Γ0 . To make this notion precise, for k = 2, . . . , 4g let Bt
= ρt (βk ) (here βk ∈ π1 is the
(k)
B0
homotopy class of loops for which ρ0 (βk ) =
as described above). Then, given = (t) > 0 the
representation ρt is said to be -close to ρ0 if for each k = 2, . . . , 4g we have
(k)
(k) Bt − B0 < measured using the l2 -norm on SU(2, 1). In the same way, for k = 1, . . . , 4g let αk be the homotopy
(k)
(k)
class of loops in π1 so that A0 = ρ0 (αk ). Then we define At = ρt (αk ).
Our goal will be to show that there exists an depending only on ρ0 so that all representations
ρt that are -close to ρ0 are complex hyperbolic-quasi-Fuchsian. In order to achieve this goal we
will construct a domain ∆t and use Theorem 4.2 to show that ∆t is a fundamental domain for
Γt = ρt (π1 ). Moreover, this will also imply that ρt is faithful, and Γt is totally loxodromic and
geometrically finite. In other words, Γt is complex hyperbolic quasi-Fuchsian.
(1)
We begin by constructing the edges of ∆t . Let Rt
plane. For k = 2, . . . , 4g we define
(k)
Rt
(5.2)
(k)
Rt
= RJ , the totally imaginary Lagrangian
to be the Lagrangian plane
(k)
= Bt
(1) Rt
(k)
= Bt (RJ ).
We will prove the following theorem in the next section.
Theorem 5.1. There exists 1 = 1 (ρ0 ) > 0 so that if < 1 then the Lagrangian planes
(1)
(4g)
Rt , . . . , Rt
are disjoint.
OPEN SETS OF MAXIMAL DIMENSION IN COMPLEX HYPERBOLIC QUASI-FUCHSIAN SPACE
(k)
(k+1)
Suppose that the disjoint Lagrangian planes R0 and R0
the side
(k)
S0 .
Then we define the corresponding side
15
are edges of ∆0 in the boundary of
(k)
St
of ∆t as follows. From Theorem 5.1 we
(k)
(k+1)
(k)
see that the Lagrangian planes Rt and Rt
are disjoint, and so determine a pack Pt . Define
(k)
(k)
(k)
(k+1)
the side St to be that part of Pt lying between Rt and Rt
. We emphasise that once we
(k)
(k)
have defined the Lagrangian planes Rt , the construction of St is canonical. Thus, since the side
(k)
(k)
pairing maps match the edges Rt they automatically match the sides St . We will prove this
theorem in the next section.
Theorem 5.2. There exists 2 = 2 (ρ0 ) with 0 < 2 < 1 so that for all < 2 we have:
(1)
(4g)
(1)
(4g)
(i) the sides St , . . . , St
only intersect in the Lagrangian planes Rt , . . . , Rt ;
(ii) the combinatorial pattern of this intersection is the same as that for the faces of ∆0 ;
(iii) there is a λ > 0 so that disjoint sides of ∆t are at least a distance λ apart.
We claim that ∆t satisfies the conditions of Poincaré’s theorem, and so is a fundamental domain
for Γt . First, observe that the following facts follow immediately from our construction:
(4j+1)
(i) For j = 0, . . . , g − 1, the map At
(4j+2)
(iv)
(v)
(vi)
(4j+3)
to the side St
(4j+2)
and At
(4j+4)
sends St
(ii)
(iii)
(4j+1)
sends the side St
to the side St
. So (S.1) is satisfied.
(4j+1)
(4j+3) −1
(4j+2)
(4j+4) −1
At
= A0
and At
= A0
so (S.2) is satisfied.
Using Theorem 5.2(i) and (ii) together with the separation properties of packs, Proposition
3.3, we see that (S.3), (S.4) and (S.5) are satisfied.
Using Theorem 5.2(iii) we see that (S.6) is satisfied.
From Theorem 5.1 we immediately obtain condition (E.1).
Again, use of Theorem 5.2 (i) and (ii) and Proposition 3.3, shows that (E.2) is satisfied.
Furthermore, there are no reflection relations and only one cycle relation:
g−1
Y
(4j+2)
At
(4j+1) −1
At
(4j+2) −1
At
(4j+1)
At
= I.
j=0
Hence ∆t satisfies the conditions of Poincaré’s theorem and thus Γt is discrete, totally loxodromic
and is geometrically finite. Moreover by (vi) above we immediately see that Γt is a faithful representation of π1 . This has proved our main theorem subject to verifying Theorems 5.1 and 5.2
which we do in the next section.
6. The Technical Results
6.1. Some preparatory results. The following simple lemmas are going to be needed below.
Lemma 6.1. Let A and B be m × m matrices. Then for each positive integer n ≥ 1,
√
(i) |tr(A)| ≤ mkAk,
(ii) kAAT − BB T k ≤ kA − Bk2 + 2kBkkA − Bk,
(iii) (A + B)n − B n ≤ (kAk + kBk)n − kBkn .
Proof. Writing A = (aij ) and B = (bij ) where i and j run from 1 to m, we have
!1
m
m
2
X
X
√
√
2
|tr(A)| ≤
|aii | ≤ m
|aii |
≤ mkAk,
i=1
where we have used the inequality (x1 + · · · + xm
i=1
)2
≤ m(x21 + · · · + x2m ).
OPEN SETS OF MAXIMAL DIMENSION IN COMPLEX HYPERBOLIC QUASI-FUCHSIAN SPACE
16
The proof of (ii) only uses standard properties of matrix norms:
kAAT − BB T k ≤ kAAT − AB T k + kAB T − BB T k
≤ kAkkA − Bk + kA − BkkBk
≤ kA − Bk + kBk kA − Bk + kA − BkkBk
= kA − Bk2 + 2kBkkA − Bk.
The proof of (iii) is by induction. The case n = 1 is obvious. Suppose now that
(A + B)n−1 − B n−1 ≤ kAk + kBk n−1 − kBkn−1 .
Then,
k(A + B)n − B n k = (A + B)n−1 (A + B) − B n−1 B = (A + B)n−1 A + (A + B)n−1 − B n−1 B ≤ kA + Bkn−1 kAk + (A + B)n−1 − B n−1 kBk
n−1
n−1
≤ kAk + kBk
kAk + kAk + kBk
− kBkn−1 kBk
n
= kAk + kBk − kBkn .
Lemma 6.2. If A and B are 3 × 3 matrices so that JA∗ J = −A and JB ∗ J = B then
kA + Bk2 = kAk2 + kBk2 .
Proof. It suffices to show that hhA, Bii = 0 and the result will follow from Pythagoras’ theorem. In
fact,
hhA, Bii = < tr(AB ∗ ) = −< tr(JA∗ BJ) = −< tr(BA∗ ) = −hhB, Aii = −hhA, Bii.
Lemma 6.3. Let U = (0, +∞) × (−π, π). For each δ + iφ ∈ U let τ : U → C be the function
τ (δ, φ) = 2 cosh(δ)e−iφ + e2iφ .
Then τ is continuously differentiable and invertible everywhere on U .
Moreover, suppose that V is a closed, convex subset of the image of U under τ . Then there exists
a constant M so that for all δ1 , δ2 , φ1 and φ2 for which τ (δ1 , φ1 ) and τ (δ2 , φ2 ) lie in V we have
|δ1 − δ2 | ≤ M τ (δ1 , φ1 ) − τ (δ2 , φ2 ),
|φ1 − φ2 | ≤ M τ (δ1 , φ1 ) − τ (δ2 , φ2 ).
Proof. It is clear that τ is continuously differentiable on the whole of U . Write
u(δ, φ) = 2 cosh(δ) cos(φ) + cos(2φ),
v(δ, φ) = −2 cosh(δ) sin(φ) + sin(2φ).
Then τ (δ, φ) = u(δ, φ) + iv(δ, φ). The Jacobian of these functions is
uδ uφ
J(u, v) = det
vδ vφ
2 sinh(δ) cos(φ) −2 cosh(δ) sin(φ) − 2 sin(2φ)
= det
−2 sinh(δ) sin(φ) −2 cosh(δ) cos(φ) + 2 cos(2φ)
= −4 sinh(δ) cosh(δ) − cos(3φ) ,
OPEN SETS OF MAXIMAL DIMENSION IN COMPLEX HYPERBOLIC QUASI-FUCHSIAN SPACE
17
which is negative on the whole of U and so τ is invertible there.
In particular, we can express δ and φ as functions δ(u, v) and φ(u, v) of u and v. Using the mean
value theorem in two variables (see for example equation (38) on page 67 of [5]) we find that when
τ1 = u1 + iv1 and τ2 = u2 + iv2 lie in a convex domain in C then there exist constants Mδ and Mφ
so that
δ(u1 , v1 ) − δ(u2 , v2 ) ≤ Mδ |u1 − u2 + iv1 − iv2 | = Mδ |τ1 − τ2 |,
φ(u1 , v1 ) − φ(u2 , v2 ) ≤ Mφ |u1 − u2 + iv1 − iv2 | = Mφ |τ1 − τ2 |.
Taking M to be the larger of Mδ and Mφ gives the result.
Lemma 6.4. Let ιJ be reflection in RJ , the standard purely imaginary Lagrangian plane. Then
ιJ B −1 ιJ = B T for any B ∈ SU(2, 1).
Proof. Writing


a b c
B = d e f 
g h j
we have
 

 



−az 1 − dz 2 − gz 3
z1
−z 1
az1 + dz2 + gz3
ιJ B −1 ιJ z2  = ιJ B −1 −z 2  = ιJ  −bz 1 − ez 2 − hz 3  =  bz1 + ez2 + hz3  = B T
z3
z3
cz1 + f z2 + jz3
cz 1 + f z 2 + jz 3
 
z1
z2  .
z3
This leads to the following result which uses Theorem 2.1 to give an algebraic criterion for when
two Lagrangian planes are disjoint.
e be any elements of SU(2, 1). The Lagrangian planes B(RJ ) and B(R
e J ) are
Lemma 6.5. Let B, B
e −1 is loxodromic, where C = BB T and C
e=B
eB
eT .
disjoint if and only if CC
Proof. The involution fixing B(RJ ) is
ι = BιJ B −1 = BB T ιJ = ιJ (B −1 )T B −1 = ιJ (BB T )−1 ,
e J ) is
where we have used Lemma 6.4. That is ι = CιJ = ιJ C −1 . Likewise the involution fixing B(R
e J = ιJ C
e −1 . Using Theorem 2.1 we see that the Lagrangian planes are disjoint if and only if
e
ι = Cι
the product of the involutions ι and e
ι is loxodromic. But
e J ιJ C −1 = CC
e −1 .
e
ιι = Cι
The result follows immediately.
6.2. The edges are disjoint. In order to prove Theorem 5.1, we must show that for each pair of
(j)
distinct j, k = 1, . . . , 4g the Lagrangian planes Rt
(j)
(j)
(k)
(k)
(j)
(k)
= Bt (RJ ) and Rt
(k)
= Bt (RJ ) are disjoint.
We know that R0 = B0 (RJ ) and R0 = B0 (RJ ) are disjoint and a distance λ0 > 0 apart. Thus
Theorem 5.1 is a consequence of the following result:
OPEN SETS OF MAXIMAL DIMENSION IN COMPLEX HYPERBOLIC QUASI-FUCHSIAN SPACE
18
e0 ∈ SO(2, 1) and that the Lagrangian planes B0 (RJ ) and
Proposition 6.6. Suppose that B0 , B
e0 (RJ ) are a distance λ0 apart. There exists > 0 so that for all Bt and B
et in SU(2, 1) with
B
et − B
e0 k < then Bt (RJ ) and B
et (RJ ) are disjoint and a distance λt apart
kBt − B0 k < and kB
e0 .
where cosh(λt ) ≥ cosh(λ0 ) − O(). Here O() is a function only depending on , B0 and B
e0 (RJ ) is C
e0 C −1
Proof. Using Lemma 6.5, we see that the product of the involutions in B0 (RJ ) and B
0
e0 = B
e0 B
e T . Thus C
e0 C −1 is loxodromic and, moreover,
where C0 = B0 B0T and C
0
0
e0 C −1 ) = 2 cosh ρ B0 (o), B
e0 (o) + 1 = 2 cosh(λ0 ) + 1
tr(C
0
et C −1 ). From the triangle
et C −1 is loxodromic. We estimate tr(C
Similarly, we must show that C
t
t
inequality,
et C −1 ) ≥ tr(C
e0 C −1 ) − tr(C
et C −1 − C
e0 C −1 )
tr(C
t
t
0
0
√ et C −1 − C
e0 C −1 ,
≥ 2 cosh(λ0 ) + 1 − 3C
t
0
where we have used Lemma 6.1 (i). However, using Lemma 6.1 (ii) we see that
et C −1 − C
e0 C −1 C
t
0
et − C
e0 Ct − C0 + C
e0 Ct − C0 + C
et − C
e0 C0 ≤ C
e0 B
et − B
e0 Bt − B0 2 + 2B0 Bt − B0 et − B
e0 2 + 2B
B
≤
2 e0 B
et − B
e0 .
et − B
e0 2 + 2B
e0 Bt − B0 2 + 2B0 Bt − B0 + B0 2 B
+B
et C −1 − C
e0 C −1 = O() and so tr(C
et C −1 ) > 3. Thus C
et C −1 is loxodromic.
Hence C
t
t
t
0
et C −1 ) = 2 cosh(λt )e−iψt + e2iψt and so
Finally, tr(C
t
e0 C −1 ) − O() = 2 cosh(λ0 ) + 1 − O()
et C −1 ) ≥ tr(C
2 cosh(λt ) + 1 ≥ tr(C
t
0
as required.
6.3. The sides of ∆t . We consider the following situation. Let B0 ∈ SO(2, 1) be a loxodromic
element of SU(2, 1). We have already seen that the Lagrangian planes RJ and B0 (RJ ) are disjoint
and C0 = B0 B0T is loxodromic. Let P0 be the pack determined by RJ and B0 (RJ ). Then, from
x/2
Proposition 3.1, we see that the slices of P0 are the Lagrangian planes C0 (RJ ) for x ∈ R. Since
1/2
B0 (RJ ) = C0 (RJ ) (see Proposition 3.1 (i) with x = 1), we see that the slices of P0 in the side S0
x/2
are the Lagrangian planes C0 (RJ ) for x ∈ [0, 1].
We want to consider the pack determined by RJ and Bt (RJ ) where Bt is in SU(2, 1) and
kBt − B0 k < . We saw in Proposition 6.6 (with B0 = I) that we may choose so that the
Lagrangian planes RJ and Bt (RJ ) are disjoint. Thus we may define Pt to be the pack Pt determined by RJ and Bt (RJ ). Writing, Ct = Bt BtT , we see that the slices of the side St contained in Pt
x/2
(RJ ) for x ∈ [0, 1], which are all disjoint.
x/2
x/2 The goal of the rest of this section will be to show that Ct − C0 = O() for x ∈ [0, 1],
which is Corollary 6.14 below. Following Lemma 2.2, we write
(6.1)
Ct = e2iφt I + sinh(δt )e−iφt Et + cosh(δt )e−iφt − e2iφt Et2 ,
(6.2)
C0 = I + sinh(δ0 )E0 + cosh(δ) − 1 E02
are the Lagrangian planes Ct
OPEN SETS OF MAXIMAL DIMENSION IN COMPLEX HYPERBOLIC QUASI-FUCHSIAN SPACE
19
for some Et , E0 in su(2, 1) with Et3 = Et , E03 = E0 . Furthermore, using Lemma 6.1 (ii),
kCt − C0 k ≤ kBt − B0 k2 + 2kB0 k kBt − B0 k = O().
Lemma 6.7. If C0 and Ct are given by (6.1) and (6.2) and kBt − B0 k < , then |δt − δ0 | = O()
and |φt | = O().
√
Proof. Write tr(Ct ) − tr(C0 ) = η. Then, η = tr(Ct − C0 ) ≤ 3kCt − C0 k = O() using Lemma
6.1 (i). We write tr(C0 ) = 2 cosh(δ0 ) + 1 = τ (δ0 , 0) and tr(Ct ) = 2 cosh(δt )e−iφt + e2iφt = τ (δt , φt )
where τ (δ, φ) is the function of Lemma 6.3. The closed η-ball centred at τ (δ0 , 0) is a convex subset
of the image of U under τ . From Lemma 6.3 there is a positive constant M so that
|δt − δ0 | ≤ M τ (δt , φt ) − τ (δ0 , φ0 ) = M tr(Ct ) − tr(C0 ) = M η = O().
Similarly |φt | = |φt − 0| ≤ M η = O() and our assertion is proved.
Lemma 6.8. If Ct and C0 are given by (6.1) and (6.2) with kCt − C0 k = O(), then kEt k is
bounded by terms involving kE0 k, δ0 and .
Proof. Writing Ct and C0 in the form of (6.1) and (6.2) we have
kCt − C0 k
= e2iφt I + sinh(δt )e−iφt Et + cosh(δt )e−iφt − e2iφt Et2 − I − sinh(δ0 )E0 − cosh(δ0 ) − 1 E02 .
Thus, using the triangle inequality,
cosh(δt )e−iφt − e2iφt kEt k2
≤ kCt − C0 k + (e2iφt − 1)I + sinh(δt )e−iφt Et − sinh(δ0 )E0 − (cosh(δ0 ) − 1)E0 ≤ sinh(δt )kEt k + kCt − C0 k + 4 sin2 (φ)kIk + sinh(δ0 )kE0 k + cosh(δ0 ) − 1 kE0 k2 .
Since kCt − C0 k, |δt − δ0 | and |φt | are all O() this shows that
sinh(δ0 )kEt k + sinh(δ0 )kE0 k + cosh(δ0 ) − 1 kE0 k2
kEt k ≤
+ O().
cosh(δ0 ) − 1
2
Therefore
kEt k ≤
sinh(δ0 )
+ kE0 k + O().
cosh(δ0 ) − 1
This proves the result.
Let Dt and D0 be defined by Ct = exp(Dt ) and C0 = exp(D0 ). Using Lemma 2.3, we see that
Dt and D0 have the form
(6.3)
Dt = 2iφt I + δt Et − 3iφt Et2 ,
D 0 = δ 0 E0 .
As in the proof of Lemma 2.3 we may decompose
Ct = I + sinh(δt )Et + cosh(δt ) − 1 Et2 e2iφt I + e−iφt Et2 − e2iφt Et2 .
We write
(6.4)
(6.5)
∆t = I + sinh(δt )Et + cosh(δt ) − 1 Et2 = exp(δt Et ),
2iφt
−iφt 2
2iφt 2
Φt = e I + e
Et − e Et = exp(2iφt I − 3iφt Et2 ).
OPEN SETS OF MAXIMAL DIMENSION IN COMPLEX HYPERBOLIC QUASI-FUCHSIAN SPACE
20
Lemma 6.9. Suppose that ∆t and Φt are given by (6.4) and (6.5) with |φt | = O(). Then
(i) kΦt − Ik = O(),
(ii) k∆t − C0 k = O().
Proof. First, we have
kΦt − Ik = (e2iφt − 1)I + (e−iφt − e2iφt )Et2 ≤ 2 sin |φt | kIk + 2 sin |3φt /2| kEt k2 = O().
This proves (i). We also have
k∆t − C0 k = (Ct − C0 ) − ∆t (Φt − I) ≤ kCt − C0 k + k∆t kkΦt − Ik = O()
where we have used the fact that k∆t k is bounded, which follows from kEt k being bounded and
|δt − δ0 | = O().
Lemma 6.10. If Et and E0 are as in (6.1) and (6.2) then
cosh(δt ) − 1 Et2 − cosh(δ0 ) − 1 E02 = O().
sinh(δt )Et − sinh(δ0 )E0 = O(),
Proof. Since JEt∗ J = −Et and JE0∗ J = −E0 we have J(Et2 )∗ J = Et2 and J(E02 )∗ J = E02 . From
Lemma 6.2, we see that
2
k∆t − C0 k2 = sinh(δt )Et − sinh(δ0 )E0 + cosh(δt ) − 1 Et2 − cosh(δ0 ) − 1 E02 2 2
= sinh(δt )Et − sinh(δ0 )E0 + cosh(δt ) − 1 Et2 − cosh(δ0 ) − 1 E02 .
Thus the result follows since k∆t − C0 k = O().
Lemma 6.11. With Et and E0 as in (6.1) and (6.2) then kδt Et − δ0 E0 k = O().
Proof.
kδt Et − δ0 E0 k ≤ |δt − δ0 | kE0 k + |δt | kEt − E0 k
≤ |δt − δ0 | kE0 k + sinh(δt ) kEt − E0 k
≤ |δt − δ0 | kE0 k + sinh(δt ) − sinh(δ0 ) kE0 k + sinh(δt )Et − sinh(δ0 )E0 .
This is O() using Lemmas 6.7 and 6.10.
Lemma 6.12. If Dt and D0 are given by (6.3) and kBt − B0 k < , then
kDt − D0 k = O().
Proof. Using Lemmas 6.7, 6.8 and 6.11 we have:
kDt − D0 k = kδt Et − δ0 E0 + 2iφt I − 3iφt Et2 k ≤ kδt Et − δ0 E0 k + |φt | k2I − 3Et2 k = O().
Now, consider the expansions
(6.6)
Ctx
= exp(xDt ) =
∞
X
xn
n=0
n!
Dtn ,
C0x
= exp(xD0 ) =
where x ∈ R and Ct and C0 have the forms (6.1) and (6.2).
∞
X
xn
n=0
n!
D0n ,
OPEN SETS OF MAXIMAL DIMENSION IN COMPLEX HYPERBOLIC QUASI-FUCHSIAN SPACE
21
Lemma 6.13. If Ctx and C0x are given by (6.6), then
kCtx − C0x k ≤ exp xkD0 k exp xkDt − D0 k − 1 .
Proof. We use (6.6) to write Ctx − C0x as an infinite series. Using the triangle inequality, Lemma
6.1 (iii) and the fact that the norm is sub-multiplicative, we have
∞
X
xn n
x
x
kCt − C0 k ≤
Dt − D0n n!
=
n=1
∞
X
n=1
∞
X
n
xn n
D
+
(D
−
D
)
−
D
0
t
0
0
n!
n
xn kD0 k + kDt − D0 k − kD0 kn
n!
n=1
= exp xkD0 k + xkDt − D0 k − exp xkD0 k .
≤
Lemma 6.13 and Lemma 6.12 immediately induce the following.
x/2
Corollary 6.14. If kBt − B0 k < and x ∈ [0, 1] then kCt
x/2
− C0 k = O().
6.4. The sides are disjoint. We consider two sides of our polyhedron ∆t . As above we may take
one of these sides to be St with edges RJ and Bt (RJ ). Writing Ct = Bt BtT , the slices of St are the
x/2
(RJ ) for x ∈ [0, 1], which are all disjoint.
We need to consider a second side. First consider the pack Pet determined by disjoint Lagrangian
Lagrangian planes Ct
e y/2 (RJ ) for y ∈ R, where C
et = B
et B
e T . The image of Pet
et (RJ ). The slices of Pet are C
planes RJ and B
t
t
b
b
b
b
e
under Bt is the pack Pt determined by the Lagrangian planes Bt (RJ ) and Bt Bt (RJ ). Its slices are
bt C
e y/2 (RJ ) for y ∈ R. If B
bt (RJ ) and B
bt B
et (RJ ) are edges of ∆t bounding a
the Lagrangian planes B
t
bt C
e y/2 (RJ ) for y ∈ [0, 1].
side Sbt contained in Pbt , then the slices of Sbt are the Lagrangian planes B
t
In order to show that the non-adjacent sides St and Sbt are disjoint, it suffices to show that each
pair of slices is disjoint. Using Lemma 6.5, this is equivalent to showing that
b T −x
bt C
e y/2 B
bt C
e y/2 T C x/2 (C x/2 )T −1 = B
bt C y B
B
t
t
t
t
t t Ct
is loxodromic (we have used CtT = Ct to show that (Ctx )T = Ctx and so on). Notice that if we apply
(j)
bt C y B
b T −x by B (j) . Thus our assumption
a further Bt to both St and Sbt , this only conjugates B
t t Ct
t
that one of the edges of St is RJ involves no loss of generality.
x/2
b0 C
e y/2 (RJ )
Consider the corresponding sides S0 and Sb0 of ∆0 . Denote their slices C (RJ ) and B
0
0
respectively. We assume that S0 and Sb0 are disjoint and so they are a distance λ0 > 0 apart.
Proposition 6.15. Suppose that S0 and Sb0 are sides of ∆0 a distance λ0 > 0 apart. There exists
et and B
bt in SU(2, 1) with kBt −B0 k < , kB
et − B
e0 k < and kB
bt − B
b0 k < ,
> 0 so that for all Bt , B
the sides St and Sbt are disjoint and a distance λt apart, where cosh(λt ) ≥ cosh(λ0 ) − O().
Proof. This will follow from Proposition 6.6 provided we can show that
x/2
x/2 bt C
e y/2 − B
b0 C
e y/2 = O().
C
− C0 = O() and B
t
t
0
OPEN SETS OF MAXIMAL DIMENSION IN COMPLEX HYPERBOLIC QUASI-FUCHSIAN SPACE
22
The first of these is Corollary 6.14. Moreover,
y/2
y/2
y/2 b0 C
e e y/2 ≤ B
bt − B
b0 C
e −C
e y/2 + B
b0 C
e −C
e y/2 + B
bt − B
e y/2 − B
b0 C
bt C
B
t
t
t
0
0
0
0
Thus the second also follows from Corollary 6.14.
We now consider a pair of adjacent sides St and Set of ∆t . Using the discussion above, we
may assume that these sides are St determined by RJ and Bt (RJ ) and Set determined by RJ and
etT , from Proposition 6.5 it suffices to show that C
e y C −x
et (RJ ). Writing Ct = Bt BtT and C
et = B
et B
B
t
t
is loxodromic for all (x, y) ∈ [0, 1] × [0, 1]. Observe that if x or y is zero then we already have
the result from Proposition 3.1 (iv). Thus it suffices to consider (x, y) ∈ (0, 1] × (0, 1]. Again, our
assumption that St and Set intersect in RJ involves no loss of generality.
The sides St and Set are deformations of sides S0 and Se0 of ∆0 , which intersect in RJ by hypothesis.
e y C −x is loxodromic for all (x, y) ∈ (0, 1] × (0, 1].
Thus we know that C
0 0
x/2
e y/2 (RJ )
If x and y are not both small then there exists λ0 > 0 so that the slices C0 (RJ ) and C
0
are a distance at least λ0 apart. Choosing x and y so that this λ0 is large compared to , we can
x/2
e y/2 (RJ ) are disjoint:
use Proposition 6.6 to show that the slices C (RJ ) and C
t
t
x/2
e y/2 (RJ ).
Proposition 6.16. Suppose that S0 and Se0 are sides of ∆0 with slices C0 (RJ ) and C
0
e
Given η > 0 there exists > 0 so that for all Bt and Bt in SU(2, 1) with kBt − B0 k < and
et − B
e0 k < then for all (x, y) ∈ (0, 1] × (0, 1] − (0, η] × (0, η] the slices C x/2 (RJ ) and
kB
t
y/2
e
e
C (RJ ) of St and St are disjoint.
t
Proof. This will again follow from Proposition 6.6. We can find λ0 (depending on η) so that for all
x/2
e y/2 (RJ ) are a distance λ0 apart.
(x, y) ∈ (0, 1] × (0, 1] − (0, η] × (0, η] the slices C0 (RJ ) and C
0
We also know from Corollary 6.14 that
x/2
x/2
x/2 e
e x/2 = O().
C
− C0 = O() and C
−C
t
t
0
Finally, when x and y are both small we need a different argument.
x/2
e y/2 (RJ ).
Proposition 6.17. Suppose that S0 and Se0 are sides of ∆0 with slices C0 (RJ ) and C
0
et in SU(2, 1) with kBt − B0 k < and
There exists η > 0 and > 0 so that for all Bt and B
et − B
e0 k < the slices C x/2 (RJ ) and C
e y/2 (RJ ) of St and Set are disjoint for all (x, y) ∈ (0, η]×(0, η].
kB
t
t
e y C −x is loxodromic for all (x, y) ∈ (0, η] × (0, η]. The result will then
Proof. We will show that C
t t
follow from Lemma 6.5.
e0 (RJ ). Then
Let δ0 be the distance between RJ and B0 (RJ ) and let δe0 that between RJ and B
tr(C0 ) = 2 cosh ρ o, B0 (o) + 1 = 2 cosh(δ0 ) + 1,
e0 ) = 2 cosh ρ o, B
e0 (o) + 1 = 2 cosh(δe0 ) + 1.
tr(C
Hence for x ∈ (0, 1] and y ∈ (0, 1] we have
tr C0x = 2 cosh(xδ0 ) + 1,
e y = 2 cosh(y δe0 ) + 1.
tr C
0
OPEN SETS OF MAXIMAL DIMENSION IN COMPLEX HYPERBOLIC QUASI-FUCHSIAN SPACE
23
x/2
x/2
e y/2 (RJ ).
e y/2 (o) = ΠR C
Let λx,y be the hyperbolic distance between C0 (o) = ΠR C0 (RJ ) and C
0
0
Also let ψ be the angle at o = ΠR (RJ ) in the hyperbolic plane between the geodesic arcs ΠR (S0 )
and ΠR (Se0 ). Then using plane hyperbolic trigonometry, see page 24 of [14], for the Lagrangian
plane RR which has curvature −1/4, we see that
cosh(λx,y /2) = cosh(xδ0 /2) cosh(y δe0 /2) − sinh(xδ0 /2) sinh(y δe0 /2) cos(ψ).
By omitting subscripts, we take the second order expansion
x
D+
1!
e+
e = I+ yD
= exp(y D)
1!
C x = exp(xD) = I +
ey
C
x2 2
D + higher order terms,
2!
y2 e 2
D + higher order terms,
2!
where the higher order terms involve multiples of x3 , respectively y 3 , and higher powers. We assume
that η is sufficiently small that we may neglect these higher order terms in what follows. Then,
e = 0, we have
since tr(D) = tr(D)
e y C −x ) = 3 +
tr(C
2
x2
e + y tr(D
e 2 ) + higher order terms
tr(D2 ) − xytr(DD)
2
2
where the higher order terms are of the form xa y b with a + b ≥ 3.
On the other hand, we have
e y C −x ) = 1 + 2 cosh(λx,y )
tr(C
0 0
= 4 cosh2 (λx,y /2) − 1
2
= 4 cosh(xδ0 /2) cosh(y δe0 /2) − sinh(xδ0 /2) sinh(y δe0 /2) cos(ψ) − 1
= 3 + x2 δ02 + y 2 δe02 − 2xyδ0 δe0 cos(ψ) + higher order terms.
where the last line was obtained by expanding into Taylor series. Again the higher order terms
are multiples of xa y b with a + b ≥ 3. We already know from (6.3) that tr(D02 ) = 2δ02 and
e 2 ) = 2δe2 . By comparing these two expressions for tr(C
e y C −x ) and equating coefficients, we
tr(D
0
0
0
0
e 0 ) = 2δ0 δe0 cos(ψ).
see that tr(D0 D
Consider the quadratic form
e 0 ) + y 2 tr(D
e 2 ).
q0 (x, y) = x2 tr(D02 ) − 2xytr(D0 D
0
Its discriminant d0 is
e 2 ) − tr2 (D0 D
e 0 ) = 4δ 2 δe2 − 4δ 2 δe2 cos2 (ψ) = 4δ 2 δe2 sin2 (ψ) > 0.
d0 = tr(D02 )tr(D
0
0 0
0 0
0 0
Thus q0 (x, y) is positive definite.
Similarly, consider
e t ) + y 2 tr(D
e 2)
qt (x, y) = x2 tr(Dt2 ) − 2xytr(Dt D
t
with discriminant
e t2 ) − tr2 (Dt D
e t ).
dt = tr(Dt2 )tr(D
e y C −x ) will be
If we can show that for small x and y that qt (x, y) is positive definite then tr(C
t t
bounded away from 3, which will prove our result. It suffices to show that |dt − d0 | ≤ O(), where
OPEN SETS OF MAXIMAL DIMENSION IN COMPLEX HYPERBOLIC QUASI-FUCHSIAN SPACE
24
O() is a positive function of . Indeed,
e t ) − tr2 (D0 D
e 0 )
e 2 ) − tr(D2 )tr(D
e 2 ) + tr2 (Dt D
|dt − d0 | ≤ tr(Dt2 )tr(D
t
0
0
e 2 ) − tr(D2 )tr(D
e 2 )
≤ tr(Dt2 )tr(D
t
0
0
e 0 ).
e t ) − tr(D0 D
e 0 )2 + 2tr(D0 D
e 0 ) tr(Dt D
e t ) − tr(D0 D
+tr(Dt D
e 0 ) = 2δ0 δe0 cos(ψ) ≤ 2δ0 δe0 . We estimate the other terms. In the first place
We have tr(D0 D
e t2 ) − tr(D02 )tr(D
e 02 ) = (2δt2 − 6φ2t )(2δet2 − 6φe2t ) − 4δ02 δe02 tr(Dt2 )tr(D
≤ 4δt2 δet2 − δ02 δe02 + 12δt2 φe2t + 12δet2 φ2t + 36φ2t φe2t
which is O() due to Lemma 6.7. On the other hand, using Lemma 6.1,
e 0 )
e t ) − tr(D0 D
tr(Dt D
√ e 0
e t − D0 D
3Dt D
≤
√ √ √ et − D
e 0 + 3D0 D
et − D
e 0 + 3Dt − D0 D
e 0 .
≤
3Dt − D0 D
Using Lemma 6.12 we see that this is also O() and thus our assertion is proved.
Combining Propositions 6.15, 6.16 and 6.17 proves Theorem 5.2.
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University of Durham, Durham DH1 3LE, England
E-mail address: [email protected],[email protected]